Theorem adddird

Description: Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	addcld.1	|- ( ph -> A e. CC )
	addcld.2	|- ( ph -> B e. CC )
	addassd.3	|- ( ph -> C e. CC )
Assertion	adddird	|- ( ph -> ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) )

Proof

Step	Hyp	Ref	Expression
1		addcld.1	|- ( ph -> A e. CC )
2		addcld.2	|- ( ph -> B e. CC )
3		addassd.3	|- ( ph -> C e. CC )
4		adddir	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) )